#8348913

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#39252	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4182 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

(1,2,7,11)(3,8,12,16)(4,5,9,14)(6,10,15,13)(17,18)(19,24)(20,21)(22,25)(23,26)(27,31)(28,29)(30,32)(35,39)(36,42)(43,45)(44,47);(1,3)(2,8)(4,6)(5,10)(7,12)(9,15)(11,16)(13,14)(17,19,23,20)(18,21,26,24)(22,27,30,28)(25,29,32,31)(33,34,38,41)(35,42,39,36)(37,40,46,48)(43,47,45,44);(3,12)(6,15)(8,16)(10,13)(17,19,23,20)(18,24,26,21)(22,28,30,27)(25,29,32,31)(33,35,38,39)(34,36,41,42)(37,45,46,43)(40,47,48,44);(1,4)(2,5)(3,6)(7,9)(8,10)(11,14)(12,15)(13,16)(17,20,23,19)(18,24,26,21)(22,28,30,27)(25,31,32,29)(33,34)(35,36)(37,40)(38,41)(39,42)(43,44)(45,47)(46,48);(1,5)(2,9)(3,13)(4,11)(6,8)(7,14)(10,12)(15,16)(17,21,23,24)(18,20,26,19)(22,29,30,31)(25,28,32,27)(33,36)(34,39)(35,41)(37,44)(38,42)(40,45)(43,48)(46,47);(1,6)(2,10)(3,4)(5,8)(7,15)(9,12)(11,13)(14,16)(17,22)(18,25)(19,27)(20,28)(21,29)(23,30)(24,31)(26,32)(33,37)(34,40)(35,43)(36,44)(38,46)(39,45)(41,48)(42,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gh,d^2=g,e^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8256109);